176 TRANSPORTATION RESEARCH RECORD 1191
Analysis of Tieback Slopes and Walls Using STABL5 and PCSTABL5
JAMES R. CARPENTER
The purpose of this study was to develop a convenient method plex problem. The stability of these structures is influenced for assessing the stability of tieback structures using the sim by, but not limited to (a) the lateral earth pressure behind plified methods of slices. The Load Distribution Method was the wall; (b) the deformation of the soil-structure system; ( c) developed to transmit the load from a row of tiebacks to the tieback characteristics (individual tieback loads, inclination, potential failure surface for use with the simplified methods horizontal spacing, overall length, size of anchor, method of of slices. The load from a row of tiebacks is assumed to form construction); and ( d) soil characteristics. The analysis of tied a uniform line load that is distributed to the potential failure back structures is further complicated by the fact that a two surface. This distribution is based on Flamant's distribution dimensional model is used to model a three-dimensional prob of stresses through a semi-infinite elastic medium. STABL5 lem. Analysis of the overall stability of tiedback structures is and PCSTABLS are limiting equilibrium slope stability pro grams that contain the Load Distribution Method routines. only one of the many considerations in the design of tiedback These programs may be used to analyze the stability of tied back structures for excavation support or slope stabilization. Because slopes for landslide stabilization, as well as to determine the many factors influence the stability of tiedback structures, and overall stability oftiedback walls. The programs consider mul because relatively little is known about these factors at the tiple rows of tiebacks, multiple tiedback structures as well as time of design, a conservative approach is often used by geo tieback loads, inclination, and length. The method developed technical engineers and design-and-build tieback contractors was found to give good results and is applicable to those prob in the design of tiedback structures. lems in which the application of a semi-infinite elastic half space Previously available methods for determining the internal may be used to model the slope conditions, and which may be and overall stability of multiple tiedback structures (1-3) often modeled using a two-dimensional analysis. This paper is a brief involved errors in the statement of the problem, or required review of some of the previously available methods for ana lyzing the overall stability of tiedback structures. In addition, that arbitrary assumptions be made to perform the calcula a discussion of the capabilities of ST ABLS and PCST ABLS is tions, usually by hand (4). In addition, it was extremely dif presented, and the development of the Load Distribution Method ficult and tedious to take into consideration nonhomogeneous is summarized. The assumptions used in the development of soil conditions and multiple tiebacks with the previously exist the Load Distribution Method are discussed, along with the ing methods of stability analysis. Therefore, there existed a implications and limitations of using Flamant's distribution. need for a convenient and logical method for determining the The effect of tieback load on the factor of safety is also pre internal and external stability of multiple tiedback retaining sented, along with recommendations concerning factors of safety structures considering nonhomogeneous soil conditions. for overall stability of tiedback slopes and walls. The purpose of this study was to develop a rational and convenient method of assessing the internal and overall sta The use of tiebacks in geotechnical engineering, transporta bility of tiedback and anchored retaining structures. As a tion, construction for support of transportation routes, con result, the Load Distribution Method (LDM) was developed struction excavations, and landslide control has increased sub for analyzing the stability of tiedback slopes and walls in con stantially within the last 10 to 15 years. As a result, the need junction with the simplified methods of slices contained in the for a reliable and practical method of analyzing the internal STABL programs. The LDM was originally programmed in and overall (external) stability of slopes and retaining walls the slope stability programs ST ABL4 and PCSTABL4 (5, 6) subjected to tieback anchor loads has become evident. and is also contained in STABLS and PCSTABLS. The Tiebacks are routinely used for both temporary and per STABLS and PCSTABLS versions retain all the capabilities manent support of excavated slopes. Tiedback or anchored and options of STABL4 and PCSTABL4. However, STAB LS retaining structures for temporary and permanent support of and PCSTABLS are the only versions of STABL with the excavations may consist of soldier piles with wood lagging, enhanced capabilities of analyzing potential failure surfaces, sheet piling, drilled concrete pile walls, or concrete diaphragm including ticdback slopes and walls, using Spencer's method walls constructed using the slurry trench method. Tiedback uf siil:t:S. Tht: ful:us ui this paper wiii be on the i.DivI cievei retaining structures for stabilization of embankments and slopes oped by the author because it is this method that is used most may be continuous along the length of a slope, as in soldier frequently. piles and wood lagging, or may be discontinuous, as in tied back drilled piers with concrete bearing pads or buttress ele ments placed on the face of the slope. CAPABILITIES OF STABLS AND PCSTABL5 The analysis of the stability of tiedback structures is a com- The STABLS and PCSTABLS programs calculate the factor STS Consultants Ltd., 2929-C Eskridge Road, Fairfax, Va. 22031. of safety against slope failure by a two-dimensional limiting Carpenter 177 equilibrium method. These programs are written in FOR with the microcomputer version PCSTABLS. The PLOTSTBL TRAN and contain routines for analyzing slopes and walls program is a BASIC program for plotting the graphical output subjected to tieback loads. PCSTABLS is the microcomputer from PCSTABLS using either a Hewlett-Packard HP-7470A version of the mainframe ST ABLS program and contains all two-pen plotter or an HP-7475A six-pen plotter. PLOTSTBL the options and capabilities of STABLS. The calculation of reads the plotted output file created by PCSTABLS, which the factor of safety against slope instability may be performed contains commands and coordinates for plotting. using (a) the Simplified Bishop method of slices, which is applicable to circular shaped failure surfaces; (b) the Simpli fied Janbu method of slices, which is applicable to failure ANALYSIS OF TIEDBACK WALLS AND SLOPES surfaces of a general shape, or; (c) Spencer's method of slices, which is applicable to surfaces having a circular or general Tiebacks tie a structure to a stable soil mass through an anchor shape. secured in the earth. The components of a typical tiedback The STAB LS and PCSTABLS slope stability programs fea retaining structure are shown in Figure 1. The anchor is attached ture unique techniques for random generation of potential to a steel tendon, which is also connected to the retaining failure surfaces for subsequent determination of the more structure. After installation of the tendon and grouting of the critical failure surfaces and their corresponding factors of safety. anchor, the tendon is stressed (pulled) to the desired load Circular, irregular, and sliding block surfaces may be gen using hydraulic jacks. This load is then locked off and per erated and analyzed using either a random search technique manently applied to the structure. The load in the steel tendon or specific input of the coordinates of a given potential failure applies a stabilizing force to the structure that is developed surface. by the anchor in the stable soil mass. Tiebacks are different The programs are capable of handling heterogeneous soil from deadman anchors in that the tieback anchor is made systems, isotropic and anisotropic soil strength parameters, through a hole drilled or driven into the soil for installation excess pore water pressure caused by shear, static ground of the tendon. water and surface water, pseudo-static earthquake loading Assumptions are made in the design of a tiedback retaining surcharge, and tieback loading. The tieback loading feature structure concerning the lateral earth pressure distribution provides for the input of horizontal or near-horizontal tieback behind the proposed wall. On the basis of the assumed lateral or line loads for analyzing the internal and overall stability of earth pressure distribution, the location and magnitude of tiedback or braced slopes and retaining walls. load applied to each tieback is determined for the internal Plotted output is provided as a visual aid to confirm the (local) stability of the wall. The structure-anchor system must correctness of problem input data. Error messages are gen be designed to resist the lateral earth pressures with a suitable erated within the program to pinpoint locations where input factor of safety (FOS). The tieback must then be designed to data are inconsistent with STABLS/PCST ABLS's input carry the computed load. The length of the tendon and anchor requirements. Free-format data input eases the task of input of the tieback must be made long enough so that the tieback file preparation, which results in a reduction of input mistakes. is beyond the area that would be disturbed by wall movements Plotting routines are provided for Calcomp-type plotters and it will not pull out of the soil mass in which it is secured. (STABLS version) and for Hewlett-Packard plotters for use Tiebacks tie a structure to a soil mass that must also be
STEEL TENDON PERMANENT WALio ____-..w i·-''; ......
"·.-,.. GROUTED ANCHOR
TEMPORARY SHEETING FIGURE 1 Components of a tiedback retaining structure. 178 TRANSPORTATION RESEARCH RECORD 1191
OflOUND SUflf ACE 0 B
---E,.
, __
FIGURE 2 Soil mass analyzed for overall stability analysis (7). both internally and externally stable. The shape of the soil some distance below the subgrade, passive resistance , P, will mass analyzed for overall stability is often taken to be wedge be mobilized at the base of the wall. The FOS with respect shaped, as shown in Figure 2. If a tiedback wall is properly to overall stability is defined as the ratio of the sum of the designed and the tiebacks have the desired capacity, the pres resisting forces to the sum of the driving forces. A typical sure on the wall and the tieback will create stabilizing internal suitable FOS for this type of analysis is 1.5 or greater (8). forces within this soil mass. The soil-structure-anchor system It is important to stress that the tieback force is an internal is then considered to be internally stable. force within the wedge and does not affect the external sta In addition to ensuring the internal stability of the soil bility of the soil mass. The tieback applies a load to the wall structure-anchor system, the overall stability of the system that pushes on the soil. The forces between the wall and the must also be checked and a suitable FOS determined. The anchor are equal and opposite, which tends to compress the determination of the FOS for any potential surface that passes soil. behind the ends of the tiebacks is considered a FOS with The Krantz method is based on the soil mass of Figure 2 respect to overall (external) stability (Figure 3A) , whereas (J). The Krantz method is based on laboratory tests and anal the FOS for any potential failure surface that passes between yses in which an external pull (force) is applied to the soil the ends of the tiebacks and the wall is considered a FOS with wedge. This method requires the calculation of an external respect to internal stability (Figure 3B). force that would be required to displace the soil wedge in If the external stability is insufficient, it may be increased which the tiebacks are anchored. The external force is taken by modifying the tieback geometry. This is usually accom as the possible tieback load. This method involves serious plished by lengthening the tiebacks. Because the loads in errors in the statement of the problem that make it inappro the tiebacks are internal forces within the soil mass wedge, priate for determining the overall stability of anchored struc they do not increase the overall stability of the system. Increas tures (4). The errors result from the fact that the method ing the load on the tiebacks or increasing the number of tie assumes that the tiebacks pull on the soil without pushing on backs will only serve to increase the internal stability of the the wall. The analysis proposed by Krantz assumes that the wedge. Because the cost per tieback increases as the length tieback force is an external force acting on the soil wedge, increases, it is desirable to determine the shortest length of whereas the tieback force is actually an internal force in this tiebacks, while providing a suitable FOS with respect to exter wedge. The tieback is in tension between the wall and its nal stability. anchor and is not related to the force required to move the For the overall stability of the soil-structure-anchor system, wedge. It is therefore inaccurate to treat the tieback force as the soil mass of Figure 2 is often analyzed. The wedge shape an external force acting on the wedge that results in mixing of Figure 2 may be used to expedite hand calculations and is internal and external forces ( 4). a simplification of the actual conditions. This soil-structure Unfortunately, the Krantz method and others based on this model forms the basis of the Krantz method. The forces tend method (2, 3) are still used today. In light of the inaccuracies ing to displace the soil mass are weight of the soil mass, W, of the methods based on the Krantz method, it is the opinion and the earth pressure, E,,, on plane, AB. The earth pressure, of the author and of Schnabel ( 4) that this type of analysis
, E0 on plane, AB, is usually taken as the active earth pressure, should be discontinued in practice because of the errors in although the at-rest earth pressure condition is sometimes the model. used, (Z). The external forces resisting displacement of the soil mass are the tangential and normal forces, F and N, on Other Uses and Methods Proposed the failure plane, AC. The failure surface, AC, may not be straight as shown, but may be curved depending on the soil Other methods have been proposed 9-11) that use a wedge parameters. In addition, if the retaining structure penetrates shaped soil mass, but properly consider the tieback force to Carpenter 179
WALL
ANCHOR WITHIN E SOIL MASS
(A) INTERNALLY STABLE BUT EXTERNALLY UNSTABLE SOIL-STRUCTURE-ANCHOR SYSTEM
POTENTIAL FAILURE SURFACE (FOS INTERNAL)
:rlEBA-CK AN CH 0 R OUTSIDE UNSTABLE SOIL MASS
(B) INTERNALLY AND EXTERNALLY STABLE SOIL-STRUCTURE-ANCHOR SYSTEM FIGURE 3 Modes of tiedback wall stability and instability. be an internal force within the soil mass. These methods are porting the soil mass directly behind the structure, must also rather straightforward for homogeneous soil conditions and apply a sufficient resisting force to the sliding mass that it is a single row of tiebacks. However, the hand calculation required intended to stabilize. Not only must the tiedback retaining for these analyses becomes cumbersome and tedious for mul wall, shown in Figure 4, be able to resist the lateral earth tiple rows of tiebacks, layered, or nonhomogeneous soil con pressure forces produced by the soil mass directly behind the ditions. In some cases the method is not able to account for wall (shaded portion), but it must also apply a resisting force these conditions or the definition of the FOS breaks down for sufficient to stabilize the sliding soil mass above the landslide purely cohesive soils. failure surface. Tiedback retaining structures are frequently used for the A sliding mass can be stabilized by increasing the resisting control of landslides. These structures, in addition to sup- forces that act on it, or by decreasing the driving forces.
LANDSLIDE RETAINING WAL FAILURE SURFACE
TIEBACKS
FIGURE 4 Landslide stabilization using tiebacks. 180 TRANSPORTATION RESEARCH RECORD 1191
Tiebacks stabilize a sliding soil mass by increasing the resisting Both components of the tieback force on the failure surface forces . Tiebacks can penetrate the sliding surface and apply tend to increase the resisting forces on the sliding mass. Any increased normal and tangential forces to the sliding body. type of stability analysis performed on slopes subjected to Tiebacks are an excellent tool for stabilizing landslides because tieback loads should consider both components of resistance they provide a force that acts in nearly an ideal direction for offered by a tiedback structure for landslide stabilization . resisting the driving forces , without seriously aggravating the stability of the slope. It can be seen from Figure 4 that the largest component of Slope Stability Program Method the tieback force acts in a horizontal direction. The tiebacks also provide a component of force to the soil mass that increases Slope stability computer programs based on a limiting equi the normal force on the sliding surface. For soils with frictional librium method of slices are routinely used for determining characteristics, it can be seen from Equation l that by increas the stability of slopes and embankments. It is therefore logical ing the effective normal force, and hence effective normal to attempt to use such an analysis tool for the determination stress , N', on the sliding surface, the resistance to sliding at of the stability of tiedback structures used for landslide con the sliding surface will be increased. trol. However, existing limiting equilibrium slope stability computer programs, with the exception of the recent versions S = c' + N' tan ' (1) of STABL , do not properly account for the presence of tie back loads in the determination of the FOS. This is especially where true for the simplified methods of slices, such as the Simplified Bishop method and the Simplified Janbu method, which do S = soil shear strength, not satisfy both force and moment (total) equilibrium (12, 13). c' effective soil cohesion, When vertical uniform distributed loads are present on the N' = effective normal stress, and crest of the slope, there are no major drawbacks to using the ' = effective angle of shearing resistance. Simplified Bishop or Janbu methods. However, when using
NORMAL STRESS DISTRIBUTION
p
" - NORMAL STRESS DISTRIBUTION
(b)
FIGURE 5 Normal stress distribution on failure surface considering a concentrated load. Carpenter 181 these methods for near-horizontal and inclined concentrated hence to nearly all slices of the sliding mass. STABLS and tieback loads, these methods are inappropriate for the fol PCST ABLS are the only known limit equilibrium slope sta lowing reasons: bility programs that attempt to account for the distribution 1. The vertical component of an inclined tieback load is of force to the failure surface caused by concentrated bound taken into account in the numerator of the FOS only on the ary loads, such as tieback loads. slice on which it acts. This does not conform to the idea that The LDM routines in STABLS/PCSTABLS are applicable stresses applied to the ground surface are diffused throughout to circular and noncircular failure surfaces and are specifically the soil mass (14-17). Shown in Figure SA is the distribution formulated to handle tieback loads, but are also capable of of normal stress on the failure surface caused by the presence handling other types of loads applied to the ground surface of a concentrated load, such as an inclined tieback load, P, such as strut loads from a braced excavation. A detailed dis applied to the face of a slope. It is clear from this figure that cussion of the derivation of LDM is beyond the scope of this the tieback load is only taken into account on the slice on paper. However, the reader may consult Carpenter (18) for which it acts. The normal stress on this slice is greatly increased, the complete derivation of the LDM. whereas nearby slices remain unaffected. As a result, the soil resistance calculated for the slice on which the tieback load acts (Equation 1) will be extremely high, whereas the soil Theory resistance of nearby slices will remain unchanged by the pres ence of the tieback load. The result is that the real FOS for A post-tensioned tieback applies a force to the structure that the slice on which the tieback load acts will be very high, it supports. This force is developed by the tieback anchor whereas the FOS of the remaining slices will be unchanged. within the soil mass. Because the forces between the wall and This problem is especially critical when the width of the slice the tieback anchor are equal and opposite, they place the soil is small, as is typically true for near-vertical tiedback retaining between the structure and the anchor in compression. structures, shown in Figure SB. The LDM diffuses the stresses caused by the tieback load 2. The horizontal component of a concentrated tieback load to the potential failure surface. This is accomplished by replac is taken into account only in the denominator of the FOS. It ing the load applied to the ground surface with a statically will be seen later that the horizontal component of the tieback equivalent distribution of forces applied to the midpoint of load produces normal and tangential forces on the base of the base of the slices along the potential failure surface. By the slices that contribute to the stability of the slope. doing so, the load is distributed to the base of all, or nearly 3. A concentrated horizontal load whose line of action passes all, of the slices, depending on the slope geometry. The dif through the center of rotation will not be taken into account fusion of stresses within the slope and the increase in forces in the FOS determination with the Simplified Bishop method along the potential failure surface are therefore considered of slices because the moment arm of the load will be zero; in the determination of the FOS. hence the resisting moment from such a load will also be zero. The distribution of stresses to the potential failure surface It is apparent from this discussion that the previously avail used in the LDM is computed according to Flamant's distri able slope stability programs using the simplified methods of bution of stresses through a semi-infinite elastic half-space slices are not capable of properly accounting for the presence (15), as proposed by Tenier and Morlier (19). Flamant's dis of concentrated loads such as tieback loads. In reality, large tribution of stresses was adapted to the problem of tiedback compressive stresses, resulting from the presence of a tieback slopes and walls because of its simplicity while evaluating and load, are distributed throughout the soil mass to the base of attempting to correct the potential errors associated with nearly all the slices of the sliding mass. These stresses cause applying this method to analysis of tiedback slopes and walls. the normal and tangential stresses to be increased on the base It is recognized that soils are not necessarily elastic and that of every slice of any failure surface that passes between the tiedback slopes or walls do not necessarily conform to a semi tieback anchor and the retaining structure. Any analysis for infinite elastic half space. Although soils do not generally determining the stability of slopes subjected to tieback loads behave as elastic materials, many solutions to the distribution should consider these increases in stresses at the base of each of stresses throughout soils have shown that this approach is slice of the sliding mass. practical and reasonable for engineering purposes (14, 16, 17). Although Flamant's formula is based on planar stress, it will be seen later that the distribution of stresses obtained using LOAD DISTRIBUTION METHOD: STABL5 AND Flamant's distribution of stresses, modified to ensure that the PCSTABL5 stress distribution obtained is in static equilibrium with the applied tieback load, seems reasonable when compared to In an attempt to account for the diffusion of compressive finite element studies performed by Tenier and Morlier (19). stresses throughout a soil mass caused by the presence of The primary assumptions used in the formulation of the tieback loads, the author has developed the LDM, for use LDM may be summarized as follows: with the simplified methods of slices. The LDM was originally programmed in the slope stability computer programs STABL4 1. It is assumed that a linearly elastic half space model may and PCSTABL4, and is also contained in the STABLS and be used to generally describe the slope conditions being ana PCSTABLS programs. The LDM eliminates the drawbacks lyzed, inherent to computerized slope stability analyses as already 2. A uniform line load is assumed to exist horizontally discussed. Unlike other slope stability programs, STABLS/ between adjacent tiebacks so that the three-dimensional tie PCSTABLS distributes the force from a concentrated load back problem may be analyzed using a two-dimensional anal throughout the soil mass to the whole failure surface and ysis, and, 182 TRANSPORTA TION RESEARCH RECORD 11 91
P =CONCENTRATED LOAD
SURFACE
LINE OF ACTION
FIGURE 6 Flamant's distribution of stress.
3. The stresses around the grouted anchor are not consid R distance to the point in question, and ered in the analysis.
. \ ~
x
POTENTIAL FAILURE SURFACE )
"---~
INCLl.L TLOAO · ~ ~·· z_ ~ - DEV - -··-·~ p
FIGURE 7 Transfer of tieback load to potential failure surface. Carpenter 183
and are anchored to horizontal load-bearing elements, such change in distribution of stress produced along the circular as steel wales that transfer the tieback load to the retaining potential failure surface shown with variation in tieback-fail structure. Based on the assumption of a uniform line load ure surface geometry. being formed between tiebacks, the LDM neglects any three The wall is shown in Figure SA, subjected to a horizontal dimensional effects that may exist. If horizontal load-bearing tieback load of 25 kips. The 25-kip tieback load is replaced members are not present on a tiedback structure, or if the by an equivalent distribution of normal and tangential stresses horizontal spacing between tiebacks is large, then this assump on the failure surface as computed using the LDM. The dis tion is no longer valid. tribution of normal stress along the failure surface is smooth The radial stress on the midpoint of the base of a given and is largest at approximately the midpoint of the failure slice is calculated using Flamant's formula: surface (point B). The same wall is shown in Figure SB, subjected to a tieback a, = 2 (TLOAD) cos (TTHETA)/('TT) (DIST) (3) load of 25 kips inclined at 30 degrees from the horizontal. The normal stress distribution is similar to that of Figure SA, where except that the stress distribution is shifted lower on the failure a, = radial stress at the midpoint of the base of surface. the slice, Similar distributions are also obtained for tangential stresses. TLOAD = equivalent tieback line load for a row of tie The stresses produced by more than one row of tiebacks are backs, superimposed to produce a combined stress on each slice. TTHETA = angle formed by the line of action of TLOAD, Tenier and Morlier (19) obtained similar distributions of and the line connecting the point of appli normal and tangential stresses. They compared the results of cation of the tieback on the ground surface finite element analyses with those obtained using Flamant's and the midpoint of the base of the slice, distribution of stresses corrected for static equilibrium. The 'TT = pi, and results verified that the distributions of normal and tangential DIST = distance between the point of application of stresses obtained using Flamant's distribution of stresses, cor TLOAD on the ground surface, and the rected for static equilibrium, were in good agreement with midpoint of the base of the slice. those obtained using a finite element model. Tenier and Mor lier's analyses were performed on simple slopes subjected to The radial force, PRAD, at the midpoint of the base of a tieback loads with homogeneous elastic soil parameters. The slice because of a given tieback load, is calculated by multi analyses did not consider nonhomogeneous soil conditions, plying the radial stress at that point in the soil mass by the ground water tables, pore pressures, or earthquake loading. length of the base of the slice (DX), see Figure 7. Because STABLS and PCSTABL5, on the other hand, are capable of of slope geometry (i.e., slope is not a semi-infinite half space), handling all the conditions already mentioned. location of the tiebacks with respect to the failure surface, and shape of the failure surface, the sum of the radial forces acting at the midpoint of the base of the slices in the direction Load Versus Factor of Safety of the line load is normally not in static equilibrium with the applied load, TLOAD. As a result, a single multiplication One of the prime factors considered in the design of tiedback factor is applied to the radial forces acting on the base of all structures is the determination of the magnitude of the applied the slices so that the sum of these forces is in equilibrium load required to ensure stability. The effect of the magnitude with the applied load. The refined radial force acting on the of the applied load was investigated for various soil conditions base of each slice is broken into its components normal and using the soil mass defined by the potential failure surface tangential to the base of each slice, PNORM and PTAN, shown in the simple slope of Figure 9. respectively. The effect of increasing the normal force on the failure The entire process outlined above is repeated for all addi surface, through the use of an applied load such as a tieback tional rows of tiebacks. The normal and tangential compo load, will not increase the mobilized soil resistance for slopes nents of the tieback loads due to all row's of tiebacks are with purely cohesive soil characteristics (Equation 1) because summed on each slice, and it is these forces that are used in
NORMAL STRESS DISTRIBUTION DUE TO 25 KIP LOAD 1 kip/SF p---1
(a) TIEBACK HORIZONTAL
. ~ ' .
kip/SF 1-----4
NORMAL STRESS DISTRIBUTION (b) TIEBACK INCLINED AT 30° FIGURE 8 Slope model for load versus FOS studies.
0 lfl N 10
0 0 ci ,, lfl +i 4- 0 lfl r-- (T) lJ) t--1 0 x 0 < lfi ('\J Y= 125 pcl 0 >- lfl f'i
0
Li J 2. 50 25. DO 37. 50 50. 00 62. 50 75. DO 87. 50 I 00. 00 X - AXIS (ft) FIGURE 9 Slope model for load versus factor of safety studies. Carpenter 185
The results obtained when the slope of Figure 9 was analyzed The analysis of the stability of tiedback structures using the with three different c - ct> soil strength characteristics are LDM is appropriate in cases in which the overall slope may shown in Figure 11. be generally modeled as a semi-infinite half space. As with purely cohesive soils, the FOS increases with The LDM does not take into account the relative stiffness increasing load. However, the rate of increase (slope of the of individual soil layers because it is based on a solution of lines) in FOS with increasing load is greater than that of Figure stress distribution through a homogeneous elastic half space. 10 for purely cohesive soil conditions. Hence, with the LDM, the stresses distributed to the potential For c - ct> soils, both components of the applied load dis failure surface are independent of the deformation charac tributed onto the failure surface act to increase stability. The teristics of the soil profile. In other words, the load from a FOS increases at a faster rate for the slope with c - ct> soil tieback will be distributed to a potential failure surface in the characteristics, because the distribution of the normal com same way for both a homogeneous soil profile and a layered ponent of the load on